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Theorem fsumrp0cl 27919
Description: Closure of a finite sum of nonnegative reals. (Contributed by Thierry Arnoux, 25-Jun-2017.)
Hypotheses
Ref Expression
fsumrp0cl.1  |-  ( ph  ->  A  e.  Fin )
fsumrp0cl.2  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  ( 0 [,) +oo ) )
Assertion
Ref Expression
fsumrp0cl  |-  ( ph  -> 
sum_ k  e.  A  B  e.  ( 0 [,) +oo ) )
Distinct variable groups:    A, k    ph, k
Allowed substitution hint:    B( k)

Proof of Theorem fsumrp0cl
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rge0ssre 11631 . . . 4  |-  ( 0 [,) +oo )  C_  RR
2 ax-resscn 9538 . . . 4  |-  RR  C_  CC
31, 2sstri 3498 . . 3  |-  ( 0 [,) +oo )  C_  CC
43a1i 11 . 2  |-  ( ph  ->  ( 0 [,) +oo )  C_  CC )
5 simprl 754 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo )
) )  ->  x  e.  ( 0 [,) +oo ) )
61, 5sseldi 3487 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo )
) )  ->  x  e.  RR )
7 simprr 755 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo )
) )  ->  y  e.  ( 0 [,) +oo ) )
81, 7sseldi 3487 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo )
) )  ->  y  e.  RR )
96, 8readdcld 9612 . . . 4  |-  ( (
ph  /\  ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo )
) )  ->  (
x  +  y )  e.  RR )
109rexrd 9632 . . 3  |-  ( (
ph  /\  ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo )
) )  ->  (
x  +  y )  e.  RR* )
11 0xr 9629 . . . . . . 7  |-  0  e.  RR*
12 pnfxr 11324 . . . . . . 7  |- +oo  e.  RR*
13 elico1 11575 . . . . . . 7  |-  ( ( 0  e.  RR*  /\ +oo  e.  RR* )  ->  (
x  e.  ( 0 [,) +oo )  <->  ( x  e.  RR*  /\  0  <_  x  /\  x  < +oo ) ) )
1411, 12, 13mp2an 670 . . . . . 6  |-  ( x  e.  ( 0 [,) +oo )  <->  ( x  e. 
RR*  /\  0  <_  x  /\  x  < +oo ) )
1514simp2bi 1010 . . . . 5  |-  ( x  e.  ( 0 [,) +oo )  ->  0  <_  x )
165, 15syl 16 . . . 4  |-  ( (
ph  /\  ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo )
) )  ->  0  <_  x )
17 elico1 11575 . . . . . . 7  |-  ( ( 0  e.  RR*  /\ +oo  e.  RR* )  ->  (
y  e.  ( 0 [,) +oo )  <->  ( y  e.  RR*  /\  0  <_ 
y  /\  y  < +oo ) ) )
1811, 12, 17mp2an 670 . . . . . 6  |-  ( y  e.  ( 0 [,) +oo )  <->  ( y  e. 
RR*  /\  0  <_  y  /\  y  < +oo ) )
1918simp2bi 1010 . . . . 5  |-  ( y  e.  ( 0 [,) +oo )  ->  0  <_ 
y )
207, 19syl 16 . . . 4  |-  ( (
ph  /\  ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo )
) )  ->  0  <_  y )
216, 8, 16, 20addge0d 10124 . . 3  |-  ( (
ph  /\  ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo )
) )  ->  0  <_  ( x  +  y ) )
22 ltpnf 11334 . . . 4  |-  ( ( x  +  y )  e.  RR  ->  (
x  +  y )  < +oo )
239, 22syl 16 . . 3  |-  ( (
ph  /\  ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo )
) )  ->  (
x  +  y )  < +oo )
24 elico1 11575 . . . 4  |-  ( ( 0  e.  RR*  /\ +oo  e.  RR* )  ->  (
( x  +  y )  e.  ( 0 [,) +oo )  <->  ( (
x  +  y )  e.  RR*  /\  0  <_  ( x  +  y )  /\  ( x  +  y )  < +oo ) ) )
2511, 12, 24mp2an 670 . . 3  |-  ( ( x  +  y )  e.  ( 0 [,) +oo )  <->  ( ( x  +  y )  e. 
RR*  /\  0  <_  ( x  +  y )  /\  ( x  +  y )  < +oo ) )
2610, 21, 23, 25syl3anbrc 1178 . 2  |-  ( (
ph  /\  ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo )
) )  ->  (
x  +  y )  e.  ( 0 [,) +oo ) )
27 fsumrp0cl.1 . 2  |-  ( ph  ->  A  e.  Fin )
28 fsumrp0cl.2 . 2  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  ( 0 [,) +oo ) )
29 0e0icopnf 11633 . . 3  |-  0  e.  ( 0 [,) +oo )
3029a1i 11 . 2  |-  ( ph  ->  0  e.  ( 0 [,) +oo ) )
314, 26, 27, 28, 30fsumcllem 13636 1  |-  ( ph  -> 
sum_ k  e.  A  B  e.  ( 0 [,) +oo ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    e. wcel 1823    C_ wss 3461   class class class wbr 4439  (class class class)co 6270   Fincfn 7509   CCcc 9479   RRcr 9480   0cc0 9481    + caddc 9484   +oocpnf 9614   RR*cxr 9616    < clt 9617    <_ cle 9618   [,)cico 11534   sum_csu 13590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-ico 11538  df-fz 11676  df-fzo 11800  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-clim 13393  df-sum 13591
This theorem is referenced by:  esumcvg  28315
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